Ground state factorization versus
frustration in spin systems
Gerardo Adesso
School of Mathematical Sciences
University of Nottingham
joint work with S. M. Giampaolo and F. Illuminati (University of Salerno)
"Hamiltonian & Gaps", 7/9/2010
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Outline
•
•
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•
•
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Spin systems and frustration
What we want to do and why
Theory of ground state factorization
Factorized solutions to frustration-free models
Frustration vs factorization and order
Summary and outlook
"Hamiltonian & Gaps", 7/9/2010
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Quantum spin systems
• N spin-1/2 particles
on regular lattices
– anisotropic
interactions of
arbitrary range
– arbitrary spatial
dimension
– translationally
invariant & PBC
– external field along z
X
1X r x x
H=
Jx Si Sl + Jyr Siy Sly + Jzr Siz Slz ¡ h
Siz
2
i;l
(
r = i - l
)
"Hamiltonian & Gaps", 7/9/2010
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Ground states
• No known exact analytical solution in general, except for
a few simple subcases (Ising, XY,...) and now a wider class
of models with nearest-neighbor interactions (see JE)
• Difficult to be determined even numerically, especially
for high-dimensional lattices (2D, 3D, ...)
• Rich phenomenology: different magnetic orderings,
critical points and quantum phase transitions
• Typically exhibit highly correlated quantum fluctuations,
i.e., they are typically entangled
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Frustration
AF
AF
AF
?
• Occurs when the ground state of the
system cannot satisfy all the couplings
• Even richer phase diagram (high
degeneracy), hence even harder to
find ground states
• In frustrated systems a magnetic order
does not freeze, which typically results
in even more correlations
• At the root of statistically fascinating phenomena and exotic
phases such as spin liquids and glasses
• Frustrated systems may play a crucial role to model high-Tc
superconductivity and certain biological processes
"Hamiltonian & Gaps", 7/9/2010
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Relevant questions
• How to define natural signatures and measures of
(classical and/or quantum) frustration?
• More generally, is it possible to tune an external field
so that a many-body model admits as exact ground
state a completely factorized (“classical-like”) state?
– This would be an instance of mean field becoming exact
• If yes, under which conditions? Does this possibility
depend on the presence of frustration? In turn, does
the fulfillment or not of this condition define a regime
of weak versus strong frustration?
"Hamiltonian & Gaps", 7/9/2010
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Ground state factorization
ground =
⊗ ⊗ ⊗ ⊗
…
• Answer to the 2° question: YES! There can exist special
points in the phase diagram of a spin system such that the
ground state is exactly a completely uncorrelated tensor
product of single-spin states: factorized ground state
• The “factorization point” is obtained for specific, finite values
of the external magnetic field (dubbed factorizing field) which
depend on the Hamiltonian parameters
• First devised by Kurmann, Thomas and Muller (1982) for 1-d
Heisenberg chains with nearest neighbor antiferromagnetic
interactions
"Hamiltonian & Gaps", 7/9/2010
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Motivations
• Many-body condensed matter perspective
– To find exact particular solutions to non-exactly solvable models
– To devise ansatz for perturbative analyses, DMRG, …
• Quantum information and technology perspective
– For several applications (e.g. quantum state transfer, dense coding,
resource engineering for one-way quantum computation), both in
the case of protocols relying on “natural” ground state
entanglement for quantum communication (in which case
factorization points should be avoided!), and for tasks which
instead require a qubit register initialized in a product state
• Statistical perspective
– To investigate the occurrence of “phase transitions in
entanglement” with no classical counterpart
– For frustrated systems: to characterize the frustration-driven
transition between order (signaled by a factorized ground state)
and disorder (landmarked by correlations in the ground state), thus
achieving a quantitative handle on the frustration degree
"Hamiltonian & Gaps", 7/9/2010
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History
• Direct method (product-state ansatz)
Analitic brute-force method, guess a product state and verify that it is the ground state
via the Schrödinger equation, becomes nontrivial for more complicate models …
– Kurmann et al. (1982): 1d Heisenberg, nearest neighbors
– Hoeger et al. (1985); Rossignoli et al. (2008): 1d Heisenberg, arbitrary
interaction range
– Dusuel & Vidal (2005): Fully connected Lipkin-Meshkov-Glick model
– Giorgi (2009): Dimerized XY chains
• Numerical method (Monte Carlo simulations)
Nightmarish for spatial dimensions bigger than two (never attempted !)
– Roscilde et al. (2004, 2005): 1d & 2d Heisenberg, nearest neighbors
"Hamiltonian & Gaps", 7/9/2010
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Our method
• Quantum informational approach
– Inspired by tools of entanglement theory
– Fully analytic method
– Requires no ansatz: the magnetic order, energy, and specific
form of the factorized ground state are obtained as a result of
the method
– Encompasses previous findings and enables the identification of
novel factorization points
– Provides self-contained necessary and sufficient conditions for
ground state factorization (in absence of frustration) in terms of
the Hamiltonian parameters
– Straightforwardly applied to cases with arbitrary range of the
interactions and arbitrary spatial dimension (e.g. cubic
Heisenberg lattices), and to systems with spatial anisotropy, etc.
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008); Phys. Rev. B 79, 224434 (2009)
"Hamiltonian & Gaps", 7/9/2010
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The ingredients /1
½k
• Under translational invariance, the ground state is completely
factorized iff the entanglement between any spin and the
block of all the remaining ones vanishes, i.e., if the marginal
(linear) entropy of a generic spin, say on site k, is zero
£ x 2
¤
y 2
2
z
• We have: SL(½k ) = 4Det½k = 1 ¡ 4 hSk i + hSk i + hSk i
so this factorization condition would depend on the
magnetizations, which are indeed the objects one cannot
compute in general models
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The ingredients /2
Uk
½k
• A generic N-qubit state is factorized iff for any qubit k there exist a unique
Hermitian, traceless, unitary operator Uk (which takes in general the form
of a linear combination of the three Pauli matrices), whose action on qubit
k leaves the global state unchanged (Giampaolo & Illuminati, 2007)
• We can define in general the “entanglement excitation energy” (EXE)
associated to spin k as the increase in energy after perturbing the system,
in its ground state, via this special local unitary Uk (Giampaolo et al., 2008)
y
In formula: ¢E(Uk ) = hªjUk HUk jªi ¡ hªjH jªi
• One can prove that, under translational invariance and under the
hypothesis [H,Sa]≠0 (a=x,y,z), the ground state is completely factorized iff
the entanglement excitation energy vanishes for any generic spin k
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The ingredients /3
1.
2.
3.
4.
A factorized ground state must have vanishing local entropy
A factorized ground state must have vanishing EXE
The ground state must minimize the energy
The Hamiltonian model H does not give rise to frustration
general theory of ground state factorization
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008); Phys. Rev. B 79, 224434 (2009)
"Hamiltonian & Gaps", 7/9/2010
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Net interactions
• All the results (form of the state, factorizing field, conditions
for ground state factorization) are only functions of the
Hamiltonian coupling parameters and of lattice geometry
factors, or more compactly, of the “net interactions”:
¥
J
A
x ,y
=
å
r
( - 1) Z r J x ,y ,
r=1
¥
J
F
x ,y
=
å
r
Z r J x ,y ,
r=1
¥
J
A ,F
x ,y
=
r
å
r=1
r
Z rJ z .
– Zr is the coordination number, i.e.
the number of spins at a distance r
from a given site
– the magnetic order is determined by
¹ = min fJxF ; JxA; JyF ; JyA g
8 F
J
>
>
< xF
Jy
¹ =
A
J
>
x
>
: A
Jy
)
)
)
)
Ferrom. order along x ;
Ferrom. order along y ;
Antiferrom. order along x ;
Antiferrom. order along y .
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. B 79, 224434 (2009)
"Hamiltonian & Gaps", 7/9/2010
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Results: Frustration-free
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. B 79, 224434 (2009)
"Hamiltonian & Gaps", 7/9/2010
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Heisenberg lattices
1D nearest neighbor
Kurmann‘82
AF (y)
AF (x)
F (x)
F (y)
• The method is versatile
and the result is totally
general: the complexity
is the same for any
spatial dimension, one
only needs to put the
correct coordination
numbers in the
definition of the net
interactions
(e.g. for nearest
neighbor models: Z=2 for
chains, Z=4 for planes,
Z=6 for cubic lattices)
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008); Phys. Rev. B 79, 224434 (2009)
"Hamiltonian & Gaps", 7/9/2010
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Other applications
• Long-range and infinite-range models
• Models with spatial anisotropy
a)
b)
c)
d)
• …
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. B 79, 224434 (2009)
"Hamiltonian & Gaps", 7/9/2010
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Frustrated systems
AF
AF
AF
• We consider a subclass of the original Hamiltonian, comprising models with
anisotropic antiferromagnetic (along x) interactions up to a maximum range rmax
• Frustration arises from the interplay between the couplings at different ranges
• We focus on 1d systems (chains) of infinite length
• For simplicity, we consider the interaction anisotropies independent on the
distance, but overall the couplings are rescaled by a range-dependent factor fr
• If all the fr’s beyond r=1 vanish, the system is not frustrated. Vice versa, if the
fr’s are all equal, the system is fully frustrated.
H=
X
i;r·rmax
x + J SySy
z z
fr (Jx Six Si+r
y i i+r + Jz Si Si+r ) ¡ h
X
Siz
i
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Short-range systems
• Simplest case: rmax = 2
(nearest and next-nearest neighbors)
– We set f1 = 1, f2 ≡ f
– The parameter f ∊[0,1] plays the role of a “frustration degree”
(a more general definition of frustration degree was given by Sen(De) et al., PRL 2008)
• Magnetic order of the ground state
f<½
standard antiferromagnet
f≥½
dimerized antiferromagnet
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Factorized ground states
• We can determine in general the form of the candidate factorized state
and the factorizing field
p
p
1 J J
(1p¡ f ) Jx Jy f < 1=2
hf =
x y =
f Jx Jy
f ¸ 1=2
2
•
The nontrivial part is now in the verification steps
– We find that for the candidate factorized state to be an eigenstate, a
necessary condition is
Jz = 0 (other possibilities lead to saturation instead of proper factorization)
– By decomposing the Hamiltonian into triplet terms, we can derive a
sufficient condition for the candidate state to be the ground state, by
testing whether its projection on three spins is the ground state of
the triplet Hamiltonian
– For frustration-free, the factorized state was always the ground state
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Factorization vs frustration
• From the triplet decomposition we find, analytically, that if the frustration
is weaker than a “critical” value, f £ f º 1 J x - J x J y + J y ,
then the ground state is factorized
ground state factorization
c
2
Jx + Jy
• The actual “compatibility
threshold” (i.e. the maximum
frustration degree that allows
ground state factorization) can
be determined numerically by
considering decompositions
into blocks of more than three
spins.
• Above this boundary the system
admits a factorized eigenstate
at h=hf, but this does not
minimize energy and instead
the ground state is entangled
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Remarks
FRUSTRATION
( h = hf )
Factorized dimerized excited eigenstate
Factorized antiferromagnetic excited eigenstate
Factorized antiferromagnetic ground state
• Frustration naturally induces correlations which tend to suppress ground
state factorization: for strong enough frustration it is not energetically
favourable for the system to arrange in a factorized state (although a
factorized state can exist in the higher-energy spectrum)
• At the factorizing field, we witness a first order quantum phase transition
(level crossing) from a factorized to an entangled ground state when the
frustration crosses the compatibility threshold
• Qualitative agreement with the results on the scaling of correlations (Sen(De)
et al., PRL 2008) and on tensor network representability (Eisert et al., PRL 2010)
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Remarks
FRUSTRATION
( h = hf )
Factorized dimerized excited eigenstate
Factorized antiferromagnetic excited eigenstate
Factorized antiferromagnetic ground state
• Reversing the perspective, we can define the regime of weak frustration as
the one compatible with ground state factorization, and the regime of
strong frustration as the one where no factorization points are allowed.
• Ground state factorization implies a definite magnetic order, thus it is a
precursor to a quantum phase transition, with critical field hc≥hf
• The regime of strong frustration is thus characterized by the fact that a
magnetic order does not freeze even at zero temperature (in layman’s
words, the ground state remains always entangled), in accordance with
other criteria to assess the frustration degree (Ramirez, Balents, ...)
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Longer-range models
• The same general features emerge by investigating frustrated
systems with interactions beyond next-nearest neighbors
– Factorized eigenstates are only allowed for Jz=0
(this limitation could be relaxed in more general non-translationally-invariant models
where the anisotropies depend individually on the distance)
– There is a compatibility threshold dividing the phase diagram into a
region of weak frustration/order/ground state factorization and a
region of strong frustration/disorder/ground state entanglement
Compatibility thresholds: Maximum value of the
frustration f as a function of the ratio Jy =Jx for
which ground state factorization points exist in frustrated antiferromagnets with rmax = 4. The black
line stands for systems with f2 = f, f3 = f=2 and
f4 = f=3, the red line for f2 = f, f3 = f=2 and
p
f4 = f=4, and the blue line for f2 = f, f3 = f= 2
and f4 = f=6.
rmax = 4
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Infinite-range models
fr
• To verify ground state factorization in fully connected models (rmax=∞),
one should decompose the Hamiltonian in terms involving n→∞ spins,
i.e., basically solve the Hamiltonian itself!
• A workaround is possible if the frustration coefficients fr follow a
decreasing functional law with r and vanish in the limit r→∞
• In this case one can impose a cutoff and deal with decompositions into
blocks of a finite number n of neighboring spins which are most effectively
coupled
• Then one takes the limit n→∞. Numerically, this means that ground state
factorization occurs if, for n large enough, the difference D between the
minimum eigenvalue m of the n-spin Hamiltonian component and the
energy associated to the candidate factorized state, vanishes
asymptotically. If r’ is the cutoff range (such that n=2r’+1), then
P
0
0
D(r ) = ¹ (r ) + 14 (Jx + Jy ) rl=1 (¡1)l fl
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Results for infinite range
• Case fr=1/r2
(weak frustration)
– The difference D(r’)=0 for any cutoff r’
 Exact factorized ground state at h f = (¼ 2 =12)
• Case fr=1/r
p
Jx Jy
(medium frustration)
– The difference D(r’) seems to converge
to 0 (more numerics needed)
 Conjectured factorized
p
ground state at h f = ln(2) J x J y
_
• Case fr=1/√r
D
(strong frustration)
– The difference D(r’) converges
to a finite value
 No factorized ground state !
S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
"Hamiltonian & Gaps", 7/9/2010
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Summary
• We approached the problem of finding exact factorized ground state
solutions to general cooperative spin models
• We devised a method to identify and fully characterize such solutions
thanks to some tools borrowed from quantum information theory
• In frustration-free systems, necessary and sufficient conditions are derived
and several novel factorized exact solutions are straightforwardly
obtained for various translationally invariant models, with interactions of
arbitrary range, and arbitrary lattice spatial dimension
• In frustrated systems, a universal behaviour emerged in which frustration
and ground state factorization are competing phenomena, the former
inducing correlations and disorder, and the latter relying on ordered,
uncorrelated magnetic arrangements. Notably short-range as well as
infinite-range (weakly) frustrated antiferromagnetic models have been
shown to admit exact factorized solutions
• Ground state factorization is an effective tool to probe quantitatively
frustrated quantum systems. The possibility vs impossibility of having a
classical-like ground state at a given value of the magnetic field defines
the regimes of weak vs strong frustration
"Hamiltonian & Gaps", 7/9/2010
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Discussion and outlook
• In this talk we only considered spin-1/2 systems, however due to a
theorem by Kurmann et al. (1982), any spin-S (S> ½) Hamiltonian which is
of the same form as a spin-1/2 Hamiltonian that admits a factorized
ground state at h=hf, will also admit a factorized ground state at the same
value of the field: greater scope of our results
• For a generic model (frustrated or not), the factorizing field (when it
exists) is a precursor to the critical field associated to a quantum phase
transition (where the external field is the order parameter), i.e. hf≤hc
• A fascinating perspective is the investigation of the ground state
entanglement structure near a factorization point: it is conjectured that
entanglement undergoes a global reshuffling and can change its typology
(demonstrated in the XY and XXZ models, Amico et al. 2007): an
“entanglement phase transition” with no classical counterpart, and
signaled by a diverging range of pairwise entanglement
• More perspectives
– Generalize the method: relax translational invariance, identify exactly
dimerized solutions, etc.
– Area laws for frustrated systems?
– Define a measure of frustration, able to distinguish quantum from classical
– ...
"Hamiltonian & Gaps", 7/9/2010
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Thank you
?
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